### Abstract

Context. In time-distance helioseismology, information about the solar interior is encoded in measurements of travel times between pairs of points on the solar surface. Travel times are deduced from the cross-covariance of the random wave field. Here, we consider travel times and also products of travel times as observables. They contain information about the statistical properties of convection in the Sun. Aims. We derive analytic formulae for the noise covariance matrix of travel times and products of travel times. Methods. The basic assumption of the model is that noise is the result of the stochastic excitation of solar waves, a random process that is stationary and Gaussian. We generalize the existing noise model by dropping the assumption of horizontal spatial homogeneity. Using a recurrence relation, we calculate the noise covariance matrices for the moments of order 4, 6, and 8 of the observed wave field, for the moments of order 2, 3 and 4 of the cross-covariance, and for the moments of order 2, 3 and 4 of the travel times. Results. All noise covariance matrices depend only on the expectation value of the cross-covariance of the observed wave field. For products of travel times, the noise covariance matrix consists of three terms proportional to 1 /T, 1 /T^{2}, and 1 /T ^{3}, where T is the duration of the observations. For typical observation times of a few hours, the term proportional to 1 /T^{2} dominates and Cov [τ_{1}τ_{2}, τ_{3}τ_{4}] ≈ Cov [τ_{1}, τ_{3}] Cov [τ_{2},τ_{4}] + Cov [τ_{1},τ_{4}] Cov [τ_{2},τ_{3}], where the τ_{i} are arbitrary travel times. This result is confirmed for p_{1} travel times by Monte Carlo simulations and comparisons with SDO/HMI observations. Conclusions. General and accurate formulae have been derived to model the noise covariance matrix of helioseismic travel times and products of travel times. These results could easily be generalized to other methods of local helioseismology, such as helioseismic holography and ring diagram analysis.

Original language | English (US) |
---|---|

Article number | A137 |

Journal | Astronomy and Astrophysics |

Volume | 567 |

DOIs | |

State | Published - Jan 1 2014 |

### Fingerprint

### Keywords

- Convection
- Methods: data analysis
- Methods: statistical
- Sun: granulation
- Sun: helioseismology
- Sun: oscillations

### ASJC Scopus subject areas

- Astronomy and Astrophysics
- Space and Planetary Science

### Cite this

*Astronomy and Astrophysics*,

*567*, [A137]. https://doi.org/10.1051/0004-6361/201423580

**Generalization of the noise model for time-distance helioseismology.** / Fournier, D.; Gizon, Laurent; Hohage, T.; Birch, A. C.

Research output: Contribution to journal › Article

*Astronomy and Astrophysics*, vol. 567, A137. https://doi.org/10.1051/0004-6361/201423580

}

TY - JOUR

T1 - Generalization of the noise model for time-distance helioseismology

AU - Fournier, D.

AU - Gizon, Laurent

AU - Hohage, T.

AU - Birch, A. C.

PY - 2014/1/1

Y1 - 2014/1/1

N2 - Context. In time-distance helioseismology, information about the solar interior is encoded in measurements of travel times between pairs of points on the solar surface. Travel times are deduced from the cross-covariance of the random wave field. Here, we consider travel times and also products of travel times as observables. They contain information about the statistical properties of convection in the Sun. Aims. We derive analytic formulae for the noise covariance matrix of travel times and products of travel times. Methods. The basic assumption of the model is that noise is the result of the stochastic excitation of solar waves, a random process that is stationary and Gaussian. We generalize the existing noise model by dropping the assumption of horizontal spatial homogeneity. Using a recurrence relation, we calculate the noise covariance matrices for the moments of order 4, 6, and 8 of the observed wave field, for the moments of order 2, 3 and 4 of the cross-covariance, and for the moments of order 2, 3 and 4 of the travel times. Results. All noise covariance matrices depend only on the expectation value of the cross-covariance of the observed wave field. For products of travel times, the noise covariance matrix consists of three terms proportional to 1 /T, 1 /T2, and 1 /T 3, where T is the duration of the observations. For typical observation times of a few hours, the term proportional to 1 /T2 dominates and Cov [τ1τ2, τ3τ4] ≈ Cov [τ1, τ3] Cov [τ2,τ4] + Cov [τ1,τ4] Cov [τ2,τ3], where the τi are arbitrary travel times. This result is confirmed for p1 travel times by Monte Carlo simulations and comparisons with SDO/HMI observations. Conclusions. General and accurate formulae have been derived to model the noise covariance matrix of helioseismic travel times and products of travel times. These results could easily be generalized to other methods of local helioseismology, such as helioseismic holography and ring diagram analysis.

AB - Context. In time-distance helioseismology, information about the solar interior is encoded in measurements of travel times between pairs of points on the solar surface. Travel times are deduced from the cross-covariance of the random wave field. Here, we consider travel times and also products of travel times as observables. They contain information about the statistical properties of convection in the Sun. Aims. We derive analytic formulae for the noise covariance matrix of travel times and products of travel times. Methods. The basic assumption of the model is that noise is the result of the stochastic excitation of solar waves, a random process that is stationary and Gaussian. We generalize the existing noise model by dropping the assumption of horizontal spatial homogeneity. Using a recurrence relation, we calculate the noise covariance matrices for the moments of order 4, 6, and 8 of the observed wave field, for the moments of order 2, 3 and 4 of the cross-covariance, and for the moments of order 2, 3 and 4 of the travel times. Results. All noise covariance matrices depend only on the expectation value of the cross-covariance of the observed wave field. For products of travel times, the noise covariance matrix consists of three terms proportional to 1 /T, 1 /T2, and 1 /T 3, where T is the duration of the observations. For typical observation times of a few hours, the term proportional to 1 /T2 dominates and Cov [τ1τ2, τ3τ4] ≈ Cov [τ1, τ3] Cov [τ2,τ4] + Cov [τ1,τ4] Cov [τ2,τ3], where the τi are arbitrary travel times. This result is confirmed for p1 travel times by Monte Carlo simulations and comparisons with SDO/HMI observations. Conclusions. General and accurate formulae have been derived to model the noise covariance matrix of helioseismic travel times and products of travel times. These results could easily be generalized to other methods of local helioseismology, such as helioseismic holography and ring diagram analysis.

KW - Convection

KW - Methods: data analysis

KW - Methods: statistical

KW - Sun: granulation

KW - Sun: helioseismology

KW - Sun: oscillations

UR - http://www.scopus.com/inward/record.url?scp=84905179195&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84905179195&partnerID=8YFLogxK

U2 - 10.1051/0004-6361/201423580

DO - 10.1051/0004-6361/201423580

M3 - Article

AN - SCOPUS:84905179195

VL - 567

JO - Astronomy and Astrophysics

JF - Astronomy and Astrophysics

SN - 0004-6361

M1 - A137

ER -